Applications of the Integral

Problems

Problems

It is not entirely obvious what is meant by the average (or mean)
value of a function on an interval. We know how to find the mean of a
finite collection of numbers (their sum divided by their number).
Needless to say, we run into problems when we want to talk about the
mean of all the values of a function on a particular interval, since
they are infinite in number.

To find our way out of this conundrum, we recall the definition of the
n
-th (upper) Riemann sum for the function
f
on the interval
[a, b]
:

Un(f, a, b) = Mi

Note that
Un(f, a, b)
is equal to to the product of
b - a
(the length
of the interval) and the mean of the values of
f
at
n
more or less
evenly-spaced points in the interval. Clearly this is a reasonable
approximate mean of the function
f
on the interval
[a, b]
.

Naturally, the same is true for the
n
th lower Riemann sum. As
n
gets larger and larger, we might imagine the upper and lower Riemann
sums to approach (one from above, one from below) the product of
b - a
and some "true" mean of the function
f
on
[a, b]
. Indeed, this
indicates precisely how we will define the average value, denoted
. We set

=

Un(f, a, b)

=

Ln(f, a, b)

=

f (x)dx

There is a way of seeing graphically that this definition makes sense. An easy
computation shows that the integral of the constant
from
a
to
b
is
equal to that of the function
f (x)
:

dx

=

|ab

=

(b - a)

=

f (x)dx

Thus,
is the height of a rectangle of length
b - a
that will have the same area as the region below the graph of
f (x)
from
a
to
b
. In physical terms, if
f (t)
represents the velocity
of a moving object, then another object moving with velocity
will travel the same distance between the moments
t = a
and
t = b
.